Series solutions of Heun-type equation in terms of orthogonal polynomials

Abstract

We introduce a nine-parameter Heun-type differential equation and obtain three classes of its solutions as series of square integrable functions written in terms of the Jacobi polynomial. The expansion coefficients of the series satisfy three-term recursion relations, which are solved in terms of orthogonal polynomials with continuous and/or discrete spectra. Some of these are wellknown polynomials while others are either new or modified versions of known ones.MSC: 34-xx, 81Qxx, 33C45, 33D45

Series Solution of a Ten-Parameter Second-Order Differential Equation with Three Regular Singularities and One Irregular Singularity

“…The recursion coefficients   , , , n n n n u w s t depend on the differential equation parameters and on n but are independent of z and such that 0 n n t s  for all n. Therefore, the solution ( ) n f z of (11) becomes a polynomial of degree n in z modulo an overall factor that depends on z but is independent of n. That is, if we write [ ( )] f z [2,18]. Thus, the solution of the differential equation (3) is equivalent to the solution of the three-term recursion relation (11). In the following, we show how to derive this three-term recursion relation.…”

“…[7][8][9][10]). More recently, we used the TRA to find solutions of the following nine-parameter Heun-type differential equation [11]     2 2 2 ( ) ( ) ( 1)( ) ( ) ( ) [12,13]. We showed that if the differential equation parameters A and B are below the critical values,…”

Open Problem in Orthogonal Polynomials

“…where  is the deformation parameter and  is a function of the parameter set . As an example, the recursion relation (2) above is obtained by deforming that of the Jacobi polynomial ( , ) [15] and [16], we also encountered modified versions of orthogonal polynomials in the Askey scheme while searching for series solutions of the following second order linear differential equation…”

Tridiagonal representation approach in quantum mechanics

“…As examples, we mention the anion problem where an electron becomes bound to a neutral molecule with an electric dipole moment [16,17], the binding of a charged particle to an electric quadrupole in two dimensions [18], energy density bands engineering [19], electric dipole and quadrupole contributions to valence electron binding in a charge-screening environment [20], etc. In applied mathematics, the TRA was also used to obtain series solutions of new types of ordinary differential equations of the second order with three and four singular points [21][22][23]. In the following section, we start by introducing and formulating the TRA and explain its two working modes.…”